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Let $X$ be a centered Gaussian random variable in a separable Hilbert space ${mathbb H}$ with covariance operator $Sigma.$ We study a problem of estimation of a smooth functional of $Sigma$ based on a sample $X_1,dots ,X_n$ of $n$ independent observations of $X.$ More specifically, we are interested in functionals of the form $langle f(Sigma), Brangle,$ where $f:{mathbb R}mapsto {mathbb R}$ is a smooth function and $B$ is a nuclear operator in ${mathbb H}.$ We prove concentration and normal approximation bounds for plug-in estimator $langle f(hat Sigma),Brangle,$ $hat Sigma:=n^{-1}sum_{j=1}^n X_jotimes X_j$ being the sample covariance based on $X_1,dots, X_n.$ These bounds show that $langle f(hat Sigma),Brangle$ is an asymptotically normal estimator of its expectation ${mathbb E}_{Sigma} langle f(hat Sigma),Brangle$ (rather than of parameter of interest $langle f(Sigma),Brangle$) with a parametric convergence rate $O(n^{-1/2})$ provided that the effective rank ${bf r}(Sigma):= frac{{bf tr}(Sigma)}{|Sigma|}$ (${rm tr}(Sigma)$ being the trace and $|Sigma|$ being the operator norm of $Sigma$) satisfies the assumption ${bf r}(Sigma)=o(n).$ At the same time, we show that the bias of this estimator is typically as large as $frac{{bf r}(Sigma)}{n}$ (which is larger than $n^{-1/2}$ if ${bf r}(Sigma)geq n^{1/2}$). In the case when ${mathbb H}$ is finite-dimensional space of dimension $d=o(n),$ we develop a method of bias reduction and construct an estimator $langle h(hat Sigma),Brangle$ of $langle f(Sigma),Brangle$ that is asymptotically normal with convergence rate $O(n^{-1/2}).$ Moreover, we study asymptotic properties of the risk of this estimator and prove minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of $langle h(hat Sigma),Brangle$ in a semi-parametric sense.
We study a problem of estimation of smooth functionals of parameter $theta $ of Gaussian shift model $$ X=theta +xi, theta in E, $$ where $E$ is a separable Banach space and $X$ is an observation of unknown vector $theta$ in Gaussian noise $xi$ with
We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_1,dots, X_n$ in a separable Hilbert space $mathbb{H}$ with unknown covariance operator $Sigma.$ The complexity of the problem is characterized by its effective
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Let $X^{(n)}$ be an observation sampled from a distribution $P_{theta}^{(n)}$ with an unknown parameter $theta,$ $theta$ being a vector in a Banach space $E$ (most often, a high-dimensional space of dimension $d$). We study the problem of estimation
For testing hypothesis on the covariance operator of functional time series, we suggest to use the full functional information and to avoid dimension reduction techniques. The limit distribution follows from the central limit theorem of the weak conv